Introduction to calculus - lesson 1

Contents

This lesson aims to provide an introduction to basic algebra. After completing this lesson, you will be able to solve simple algebraic equations using addition and subtraction. Furthermore, you will have a basic understanding of the importance of algebra as the basis for all of mathematics.

The objective of algebra is to be able to apply basic properties of mathematics to an equation, in order to obtain a useful result. Believe it or not, you have already solved many algebraic equations:

1+1=x{\displaystyle 1+1=x}

Everybody knows that the value of x{\displaystyle x} is 2{\displaystyle 2}. In order to find the value of x{\displaystyle x}, all you did was use basic addition to combine the two 1{\displaystyle 1}'s on the left-hand side of the equals sign. That's all there is to basic algebra. We want to develop methods for manipulating equations in order to solve for a variable, such as x{\displaystyle x}.

The symbol x{\displaystyle x} simply represents something that is unknown. When we initially look at the equation 1+1=x{\displaystyle 1+1=x}, we do not know what the value of x{\displaystyle x} is. Again, the point of algebra is to figure out what the value of this unknown symbol is. Since the value of x{\displaystyle x} can vary from equation to equation, x{\displaystyle x} is referred to as a variable.

Let's look at an example of an equation that you may be unfamiliar with:

1+x=2{\displaystyle 1+x=2}

It would appear that our basic mathematical methods (addition, subtraction, multiplication, or division) cannot be applied to solve this equation, and discover the value of x{\displaystyle x}. However, one of the most important properties of an algebraic equation is that you can do anything to the equation, as long as you do the same thing to both sides of the equation. What this means for our current equation, is that we can start performing operations on both sides of the equation in hopes that we can put the equation in a form where x{\displaystyle x} is by itself on one side of the equation, and the value of x{\displaystyle x} is on the other side of the equals sign.

In order to solve the current equation, we need to figure out how to get x{\displaystyle x} by itself. If we subtract 1{\displaystyle 1} from the left-hand side, we can see that, 1+x−1{\displaystyle 1+x-1}, will leave us with x{\displaystyle x} on the left of the equals sign. Remember, that if we subtract 1{\displaystyle 1} from the left-side, we must do the same on the right side. If we don't remember to do this, the equals sign doesn't have any meaning, because the two sides are no longer the same. So, subtracting 1{\displaystyle 1} from both sides leaves us with:

1+x−1=2−1{\displaystyle 1+x-1=2-1}

We are now able to perform our basic mathematical methods, namely subtraction, to simplify each side of the equation:

x=1{\displaystyle x=1}

We now have a value for x{\displaystyle x}, and thus have solved the algebraic equation for x{\displaystyle x}.

You will notice that this is the only value for x{\displaystyle x} that will cause this equation to be true. This is a very powerful result, and is the basis for all mathematics. I hope you can see how algebraic equations can become very complicated, yet manageable. In the next lesson, I will show how multiplication and division can be involved in solving algebraic equations, in much the same way as addition and subtraction. Sample problems and solutions will follow on the course's website.